1. This work grew from a study of the conditions under which a Gaussian stochastic process has a "smooth" local time for almost all sample functions [l]-[4]. It is shown here that the main calculation in our previous work involves a property of Gaussian processes which is of independent interest-local nondeterminism. Let X(t\ -oo < t < oo, be a Gaussian process with mean 0, and J an open interval on the t-axis. Suppose thatfor all s and t in J.For arbitrary t x < • • • < t mi where tjeJ, form the ratio V m of the conditional to the unconditional variance : This is a local version of the classical notion of nondeterminism: it signifies that an observation is "relatively unpredictable" on the basis of a finite set of observations from the immediate past. We find conditions under which the members of certain classes of Gaussian processes are locally nondeterministic: for example, processes of multiplicity 1, processes with stationary increments, and others.2. Local nondeterminism means that there is an unremovable element of "noise" in the local evolution of the sample function. We expect such a function to be "locally irregular". And so it is: We show that local nondeterminism is one of the two main sufficient conditions in our result AMS (MOS) subject classifications (1970). Primary 60G10, 60G15, 60G17; Secondary 60G25.
Definition 2.1. When F(x) is absolutely continuous, its derivative, denoted by >(x), is called the local time of f(t). (By "derivative" we mean the Radon-Nikodym derivative.) The characteristic function of F is the main instrument used to probe the local time. The following lemma on characteristic functions is fundamental for our study: Lemma 2.1. Let F be a distribution function whose characteristic function is square-integrable ; then F is absolutely continuous and its derivative is square-integrable. Proof. For every square-integrable function g on the real line, let g be its Fourier transform. Define the linear functional S(g), g e L2, as S(g)= f g(u)4(-u)du,
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